Probability Distribution of Discrete Random Variables
44 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is a characteristic of a discrete random variable?

  • It can assume a countable number of values. (correct)
  • It can assume values from a continuum of numbers.
  • It can assume an uncountable number of values.
  • It represents outcomes that are derived from deterministic processes.

    • Which of the following describes the probability distribution of a discrete random variable?

  • It defines probabilities that can be negative.
  • It lists probabilities for each possible outcome. (correct)
  • It assumes the sum of probabilities is less than 1.
  • It associates probabilities with continuous outcomes.

    • In a random experiment, what is the definition of a random variable?

  • A constant that describes the average outcome.
  • A variable that assumes numerical values based on random outcomes. (correct)
  • A predetermined value for each experiment.
  • A variable summarizing the deterministic outcome.

    • What would be considered a discrete random variable?

    <p>The number of students in a classroom.</p> Signup and view all the answers

    What is true about the probabilities in a probability distribution for a discrete random variable?

    <p>The sum of probabilities must equal 1.</p> Signup and view all the answers

    Which of the following is an example of a discrete random variable?

    <p>The number of cars passing through a toll booth in an hour.</p> Signup and view all the answers

    What is a defining feature of a continuous random variable?

    <p>It can assume values across intervals on the number line.</p> Signup and view all the answers

    What does p(x) represent in the context of a probability distribution for a discrete random variable?

    <p>The probability associated with a specific outcome x.</p> Signup and view all the answers

    What is the probability of observing 0 new cases of West Nile Virus in New England in a month?

    <p>0.135</p> Signup and view all the answers

    What is the probability of observing exactly 2 new cases of West Nile Virus?

    <p>0.27</p> Signup and view all the answers

    If the rate of new cases is 2 per month, what is the formula for calculating the probability of observing X cases?

    <p>P(X) = 2^X e^{-2} / X!</p> Signup and view all the answers

    As the number of observed cases increases, what generally happens to the probabilities of observing specific counts in a Poisson distribution?

    <p>They decrease to zero.</p> Signup and view all the answers

    What is the Poisson distribution primarily used to describe?

    <p>Discrete occurrences over an interval</p> Signup and view all the answers

    Which of the following is a characteristic of the Poisson distribution?

    <p>Describes rare events with a low probability of success</p> Signup and view all the answers

    What does the symbol λ (lambda) represent in the Poisson distribution?

    <p>The rate at which events occur</p> Signup and view all the answers

    Which of the following best describes the shape of a Poisson distribution curve when λ increases?

    <p>It approaches a normal distribution shape</p> Signup and view all the answers

    What type of events does the Poisson distribution not effectively model?

    <p>Frequent events with high probabilities</p> Signup and view all the answers

    In the Poisson formula, what does the term $e$ represent?

    <p>The base of the natural logarithm</p> Signup and view all the answers

    What is the relationship between the mean (μ) and the variance (σ²) in a Poisson distribution?

    <p>Mean and variance are equal</p> Signup and view all the answers

    How does the distribution of occurrences change as λ increases from low values to high values?

    <p>It becomes symmetric and unimodal</p> Signup and view all the answers

    What is the formula for the variance of a discrete random variable x?

    <p>$E[(x - u)^2]$</p> Signup and view all the answers

    According to Chebyshev's Rule, what is the minimum probability that a random variable x falls within one standard deviation of the mean?

    <p>0</p> Signup and view all the answers

    What is the expected probability of obtaining heads when flipping a fair coin?

    <p>0.5</p> Signup and view all the answers

    In a binomial distribution, what condition must be satisfied regarding the trials?

    <p>Each trial must have the same probability of success</p> Signup and view all the answers

    What is the probability of getting exactly one tail when tossing two coins?

    <p>0.5</p> Signup and view all the answers

    According to the Empirical Rule, what percentage of the data will fall within three standard deviations of the mean in a normal distribution?

    <p>99.7%</p> Signup and view all the answers

    What characterization can be made about the outcomes in a binary experiment?

    <p>There are only two possible outcomes for each trial</p> Signup and view all the answers

    If the probability of success in a trial is given as p, what is the probability of failure?

    <p>1 - p</p> Signup and view all the answers

    What is the probability that none of the selected students suffer from math anxiety?

    <p>0.16</p> Signup and view all the answers

    How is the expected value of a discrete random variable calculated?

    <p>E(x) = Σ x P(x)</p> Signup and view all the answers

    Which of the following represents the variance of a discrete random variable?

    <p>σ² = Σ (x - μ)² P(x)</p> Signup and view all the answers

    If two students are selected, what is the cumulative probability that at least one of them suffers from math anxiety?

    <p>0.84</p> Signup and view all the answers

    What does the standard deviation of a discrete random variable represent?

    <p>The variability of the variable around the mean</p> Signup and view all the answers

    In the context of the provided data, which of the following statements is true?

    <p>P(x=1) is equal to the sum of probabilities for P(NM) and P(MN).</p> Signup and view all the answers

    What is the formula to calculate the expected value (E(x)) for a discrete random variable?

    <p>E(x) = Σ x P(x)</p> Signup and view all the answers

    Which event describes the situation where a selected student does not suffer from math anxiety?

    <p>Event N</p> Signup and view all the answers

    What is the probability of seeing 5 striped trout in the next 100 yards, given an average of 3 striped trout per 100 yards?

    <p>0.1008</p> Signup and view all the answers

    Which of the following is the correct formula to calculate the probability of observing x occurrences in a Poisson distribution?

    <p>P(x) = rac{e^{- u} u^x}{x!}</p> Signup and view all the answers

    If the average number of striped trout is increased to 5 per 100 yards, what would be the new probability of seeing exactly 5 striped trout?

    <p>0.2079</p> Signup and view all the answers

    In a discrete random variable table, which variable is typically represented in the rows?

    <p>The random variable outcomes</p> Signup and view all the answers

    What is the value of P(X=0) for an average of 3 striped trout per 100 yards?

    <p>0.6065</p> Signup and view all the answers

    Which option does NOT accurately reflect an essential property of discrete random variables?

    <p>They can represent continuous outcomes.</p> Signup and view all the answers

    Identifying the correct distribution type for modeling the number of striped trout in 100 yards, which statement is true?

    <p>It is modeled by a Poisson distribution.</p> Signup and view all the answers

    For what value of $x$ does the Poisson distribution become negligible when the average rate is set to 3?

    <p>Greater than 10</p> Signup and view all the answers

    More Like This

    Use Quizgecko on...
    Browser
    Open
    Browser
    Browser